The quantities determining mechanical response of materials or objects to external force are generally referred as elastic constants. Measurements of elastic response can be performed statically and dynamically. In static measurement the deformation of material at the instance of application of external force indicates compliance of the materials. The amount of deformation is determined by elastic modulus that is a fundamental property of the material. Dynamic measurements, on the other hand, reflect the deformation that changes as a function of time for a constant applied force or frequency dependence of the deformation to external forces. Dynamic measurements produce a complex elastic modulus that changes as a function of frequency of the applied force. The real part of the complex modulus represents instantaneous response of the materials to external frequencies; the imaginary part, which is also frequency dependent, reflect viscous or anelastic response of the materials. For many materials, in particular polymers and biologic materials, the imaginary part of the complex modulus plays a dominant role in determining mechanical response of the material. Dynamic mechanical measurements, therefore, play an essential role for these families of materials.
The procedures and techniques to measure dynamic mechanical qualities of a large sample in response to a given force are well known. For example, the process for determining how a given material responds to an applied stress generally involves placing each end of a relatively large sample that is centimeters long and at least millimeters in diameter into a chuck or clamping device having sensors that can measure the amount of deformation of the sample and the forces applied to the same sample. The sensors can determine the dynamic response of the sample in response to a change in temperature or a change in magnitude and frequency of a tensile or compressive force applied to the sample. Examples of such large-scale dynamic mechanical analyzers are shown in U.S. Pat. Nos. 4,295,374, 4,418,573, 4,506,547, 5,139,079, 5,710,426 and 6,146,013.
While large-scale dynamic mechanical analysis is useful in many contexts, it suffers from a fundamental limitation due to the relatively large size required for the sample. Because the measurements of the dynamic response include contributions from the entire sample, it is not possible to isolate localized variations due to complicated fine structures or defects within a larger structure. Large-scale dynamic mechanical analysis also cannot be applied to micro and nanometer size samples or to thin films because the techniques do not work with relatively smaller sized samples. Furthermore, conventional dynamic mechanical analysis systems are subject to an upper frequency limit in the form of fundamental resonance frequency that is only a few hundred hertz due to the design volume required by such systems in order to cope with large samples. Dynamic mechanical measurements are generally performed below the initial resonant frequency of the measurement system. Because almost all large sized measurement systems have a resonance frequency of only a few hundred hertz, the effective limit on the frequency range is below 200 hz, even in well-refined systems. Dynamic mechanical measurements of materials above a kilohertz are not possible with existing measurement systems.
As a result, research has been devoted to developing instruments and techniques that are capable of performing dynamic nanomechanical analysis on samples in the micron range and smaller, and on thin films. Most of these instruments use a scanning probe microscope or atomic force microscope (AFM) as the basic instrumentation platform. The AFM has an extremely small stylus with a probe tip measuring only nanometers radius that is scanned back and forth across a sample. AFM based measurements have removed some of the constraints of large sample size for mechanical analysis and have made it possible to probe local mechanical properties. A sensor arrangement detects the movement of the stylus relative to the surface of the sample and various calculations and other measurements are used to deduce features of the sample. Examples of current instruments and techniques for performing dynamic nanomechanical analysis are shown in U.S. Pat. Nos. 5,329,808, 5,700,953, 5,764,068, 5,675,075, 5,866,807, 6,006,593, 6,095,679, 6,185,991, 6,349,591 and 6,200,022. All of the existing arts described in these patents use a piezo actuator to shake the sample or to shake the base of the cantilever as the tip of the cantilever is in contact with the sample.
The basic parameters acquired in the dynamic mechanical analysis are real and imaginary part of the modulus. The ratio of these two parameters determines the quality factor of the material, which is also interpreted and measured as phase lag between the applied stress and the responding strain of the sample. Frequency dependence of this phase lag has become the signature of viscoelasticity or anelasticity of the polymeric, biological or metallic materials. A paramount issue in dynamic mechanical analysis (DMA) instrument design is to eliminate any system mechanical resonance so that the phase lag detected does not contain any contribution of the resonance induced phase shift.
A review of the mathematical theory for measuring dynamic mechanical responses highlights the apparent reasons for this fundamental limitation. In order to determine an elastic response of a material, for example, a periodic force is applied to the sample. The elastic part of the deformation of the sample exhibits its resistance to the periodic force as the real number part of the modulus of elasticity (Ereal). The anelastic or viscoelastic deformation of the sample in response to the periodic force appears as the imaginary part of modulus of elasticity (Eimaginary). The phase lag φ measured between the applied stress represented by the periodic force and the sample response is an intrinsic material property called internal friction (Q−1), which is expressed as:Q.−1=Eimaginary/Ereal  (1)If the periodic force is applied to the sample at a frequency far below any resonant frequency or frequencies of the measurement equipment, then the internal friction (Q−1) is related to the phase lag φ between applied stress and sample deformation as:Q−1=tan φ  (2)
In mechanical measurement systems where the system exhibits a resonance, the phase lag θ between the applied stress and the measured deformation response can be induced by both the material property of the sample and the resonance of the measurement system. The relationship for this is shown in the following equation:tan θ=ωr2*Q−1/.(ωr2−ω2)  (3)where, ωr represents the resonance frequency of the measurement system and ω is the measurement force frequency of the applied stress. It can seen from Eq. (3) that only when the measurement force frequency ω is far away from the resonance frequency of the measurement system can the phase lag θ between the applied stress and the measured response be correlated to the intrinsic material property Q−1, or tan θ=Q−1 when ω<<ωr. Around the resonance point, the term 1/.(ωr2−ω2) dominates the phase between the applied stress (in the form of the reference signal from a spectra analyzer) and the measured response (in the form of a signal from the AFM), giving a 180-degree phase shift. Because the phase shift due to resonance substantially overrides any phase lag due to the intrinsic material property, the phase lag θ no longer correlates directly to mechanical properties of the sample materials, but rather is dominated by the resonance of the measurement system.
In the context of the mechanical spectra, the real part elastic modulus (Er) and imaginary part elastic modulus (Ei) of a material can be generally expressed as:Er(ω)=Er+ΣδEi*(ωτj)2/(1+(ωτj)2)  (4)andEi(ω)=ΣδEi*(ωτj)/(1+(ωτj)2)  (5)where, Er(ω) and Ei(ω) are the mechanical spectra of a viscoelastic or anelastic material, exhibiting the dependency of real and imaginary parts of the modulus of the material as a function of frequency. The summing with subscript j represents contribution of independent microscopic mechanisms to the total spectra. This frequency dependency can be measured as function of the phase lag φ according to Eq. 2. Again, Eq. (3), with respect to the phase lag θ, indicates that these measurements should be made only at frequencies far away from any mechanical resonance of the measurement system.
As a result, all of the commercially available systems capable of performing dynamical mechanical analysis (DMA) operate only at frequencies below 200 Hz because the measurement system resonance can start as low as a few hundred hertz and becomes significantly worse at higher frequencies. In the context of dynamic nanomechanical analysis using an AFM, the complications of resonance frequency of the measurement system can be even more severe due to the more complicated nature of the measurement systems. U.S. Pat. No. 5,700,953 describes the use of a small sinusoidal force imposed on the sample by a piezoelectric element that controls the distance between the tip and the sample in an effort to deal with the problems of resonance. U.S. Pat. No. 6,200,022 describes the use of small constant frequency variations in the temperature of the probe and sample that are used to create a thermal stress applied to the sample to deal with the problem of resonance in a thermal context.
In other uses of an AFM, it is common to vibrate the mounts of the probe tip against the sample at a constant frequency in order to produce intermittent contact with the sample as part of the surface measurement. This technique, known as a Tapping-Mode® AFM, was pioneered by the assignee of the present invention and is described, for example, in U.S. Pat. Nos. 5,412,980, 5,519,212 and 6,008,489. Unlike the use of an AFM for dynamic nanomechanical analysis, however, the intermittent contact mode is intentionally operated at the resonance frequency of the AFM. For example, U.S. Pat. Nos. 5,902,928 and 6,318,159, describe how to compensate for the problem of varying resonance frequency that changes as the probe tip is advanced toward a sample.
In U.S. Pat. No. 5,866,807, a two-stage approach is used to avoid the problem of resonance in measuring the dynamic mechanical deformation properties of a material using an AFM. In the technique taught by this patent, a permanent indentation is made in the material during an initial pass. The depth and the feature characteristics of that indention are then measured during a subsequent pass where the AFM is operated at the resonance frequency. While this technique offers less information because it does not apply a force signal at higher frequencies, it does avoid the problem of resonance of the AFM as a measurement system.
U.S. Pat. No. Re. 36,488 describes an embodiment of the operation of an AFM in an intermittent contact mode where changes in which the phase and frequency response of the cantilever probe over a narrow range of frequency modulations centered about the resonance frequency are measured in order to determine the degree of damping induced on the cantilever probe by the sample. While some useful information can be discerned from these measurements, again the focus of using the AFM as a measurement instrument is on operation at or near the resonance frequency. The AFM system described in this patent is not able to provide material properties in a wide range of continuous frequencies.
U.S. Pat. No. 6,318,159 describes a scanning force microscope that operates in an approach mode as a vibrating probe tip is driven into proximity with a surface of a sample to be examined, and in a scanning mode as the vibrating probe tip is moved along the surface to measure its characteristics. In the approach mode, the frequency of tip vibration is preferably varied to maintain a constant phase angle between the motion of the probe tip and the motion of the actuator driving the probe tip in vibration through a cantilever. Operation in the scanning mode follows operation in the approach mode, with vibration of the probe tip in the scanning mode being at the last frequency used in the approach mode. This patent also discloses a method for automatic engagement of the tip of a scanning force microscope, in an AC detection mode, with the probe tip being vibrated at or near its resonant frequency by an excitation voltage signal applied to the excitation segment. One component of this method modulates the tip to surface distance in order to facilitate the measurement of a value of the slope (dA/dZ) of the curve of vibration amplitude (A) as a function of the tip-to-surface separation, a value which is used as a parameter for stopping the process of increasing the engagement of the tip to the sample surface. A second component of this method uses a phase-frequency servo algorithm to monitor the shift in resonance frequency and, accordingly, to adjust the driving frequency in real time, keeping the driving frequency in a desired region of operation at all times, somewhat like a dithering signal that corrects for drive frequency in response to cantilever resonance frequency as the tip is interacting with the surface of the sample in a ‘non-contact’ mode. While this patent discloses several approaches for controlling the frequency of operation, it will be noted that the cantilever in this system is operating only at its resonance frequency.
The most used method to derive material properties on AFM platform is so called force modulation measurement. In this technique, the probe tip is brought into constant contact with sample surfaces. An AFM feedback loop using the cantilever deflection as feedback parameter is employed to maintain a constant contact force between the tip and the surfaces. A piezo element mounted on the sample holder or the tip holder is used to modulate the sample or cantilever at a fixed frequency. The phase or amplitude of the cantilever motion thus determined is then correlated to material mechanical properties. This force modulation technique can be found in a rich collection of published works or patents. However, all of the existing techniques share the same problem, i.e., the drive force used to modulate the sample or the cantilever is tens of thousands to hundreds of thousands times more than what is needed to deform the cantilever at the desired frequency. Such substantially overpowered drive creates numerous parasitic resonances, rendering wide frequency mechanical spectra measurement impossible. All the reported measurements or patents are either restricted in a single modulation frequency or a very narrow frequency range. Wide, continuous frequency range dynamic mechanical analysis of mesoscale to nanometer scale remains a challenge.
Additional problems encountered in dynamic nanomechanical analysis are the limitations imposed by the actuating materials and the mechanical bandwidth of the mechanical assembly. The bi-layer actuation generated by the piezo stack actuator, for example, currently has a mechanical bandwidth of up to 50 kHz. due to the resonance of the active layer. The piezo stack actuator also has residual or parasitic resonances spread all over the frequency range, even though the individual plates can have resonances above the MHz range.
It would be desirable to provide for a system capable of performing dynamic nanomechanical analysis over a broad range of frequencies that can overcome the limitations of existing DMA systems and that includes frequencies higher than the resonance frequency of those DMA systems.